# A triangle has sides A, B, and C. Sides A and B are of lengths #5# and #2#, respectively, and the angle between A and B is #(7pi)/12 #. What is the length of side C?

Given:

The Law of Cosines is:

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Using the law of cosines, the length of side C (c) can be calculated using the formula:

c = √(a^2 + b^2 - 2ab cos(C))

Where: a = length of side A b = length of side B C = angle between sides A and B

Given: a = 5 b = 2 C = (7π)/12

Plugging in the values:

c = √(5^2 + 2^2 - 2 * 5 * 2 * cos((7π)/12))

c ≈ √(25 + 4 - 20cos((7π)/12))

Now, compute the value of cos((7π)/12):

cos((7π)/12) ≈ -0.5176

Substitute this value into the equation:

c ≈ √(25 + 4 - 20 * (-0.5176))

c ≈ √(25 + 4 + 10.352)

c ≈ √39.352

c ≈ 6.276

Therefore, the length of side C is approximately 6.276.

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